[N,k,chi] = [225,2,Mod(46,225)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(225, base_ring=CyclotomicField(10))
chi = DirichletCharacter(H, H._module([0, 6]))
N = Newforms(chi, 2, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("225.46");
S:= CuspForms(chi, 2);
N := Newforms(S);
Newform invariants
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Character values
We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/225\mathbb{Z}\right)^\times\).
\(n\)
\(101\)
\(127\)
\(\chi(n)\)
\(1\)
\(-\beta_{8}\)
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
Refresh table
This newform subspace can be constructed as the kernel of the linear operator
\( T_{2}^{12} + 8 T_{2}^{10} + 3 T_{2}^{9} + 34 T_{2}^{8} + 8 T_{2}^{7} + 91 T_{2}^{6} + 96 T_{2}^{5} + 852 T_{2}^{4} + 321 T_{2}^{3} + 96 T_{2}^{2} + 14 T_{2} + 1 \)
T2^12 + 8*T2^10 + 3*T2^9 + 34*T2^8 + 8*T2^7 + 91*T2^6 + 96*T2^5 + 852*T2^4 + 321*T2^3 + 96*T2^2 + 14*T2 + 1
acting on \(S_{2}^{\mathrm{new}}(225, [\chi])\).
$p$
$F_p(T)$
$2$
\( T^{12} + 8 T^{10} + 3 T^{9} + 34 T^{8} + \cdots + 1 \)
T^12 + 8*T^10 + 3*T^9 + 34*T^8 + 8*T^7 + 91*T^6 + 96*T^5 + 852*T^4 + 321*T^3 + 96*T^2 + 14*T + 1
$3$
\( T^{12} \)
T^12
$5$
\( T^{12} - 6 T^{11} + 16 T^{10} + \cdots + 15625 \)
T^12 - 6*T^11 + 16*T^10 - 21*T^9 - 4*T^8 + 100*T^7 - 285*T^6 + 500*T^5 - 100*T^4 - 2625*T^3 + 10000*T^2 - 18750*T + 15625
$7$
\( (T^{6} + 6 T^{5} + 4 T^{4} - 25 T^{3} + \cdots + 20)^{2} \)
(T^6 + 6*T^5 + 4*T^4 - 25*T^3 - 25*T^2 + 20*T + 20)^2
$11$
\( T^{12} - 4 T^{11} + 11 T^{10} + \cdots + 59536 \)
T^12 - 4*T^11 + 11*T^10 - T^9 + 212*T^8 - 251*T^7 + 3901*T^6 - 3553*T^5 + 23099*T^4 - 37838*T^3 + 46448*T^2 - 19520*T + 59536
$13$
\( T^{12} + 2 T^{11} + 19 T^{10} + \cdots + 10201 \)
T^12 + 2*T^11 + 19*T^10 - 18*T^9 + 457*T^8 + 1828*T^7 + 18329*T^6 + 29186*T^5 + 194054*T^4 + 96364*T^3 + 926492*T^2 - 157560*T + 10201
$17$
\( T^{12} - T^{11} + 29 T^{10} + \cdots + 21520321 \)
T^12 - T^11 + 29*T^10 + 160*T^9 + 760*T^8 + 1159*T^7 + 44941*T^6 + 127031*T^5 + 640270*T^4 + 3548020*T^3 + 16136289*T^2 + 15304061*T + 21520321
$19$
\( T^{12} - 7 T^{11} + 73 T^{10} + \cdots + 144400 \)
T^12 - 7*T^11 + 73*T^10 - 179*T^9 + 710*T^8 - 139*T^7 + 10923*T^6 - 85007*T^5 + 311101*T^4 - 366020*T^3 + 244440*T^2 - 125400*T + 144400
$23$
\( T^{12} + 19 T^{11} + 156 T^{10} + \cdots + 4080400 \)
T^12 + 19*T^11 + 156*T^10 + 509*T^9 + 1551*T^8 + 8265*T^7 + 35410*T^6 - 52695*T^5 + 559235*T^4 + 656950*T^3 + 3413700*T^2 + 2403800*T + 4080400
$29$
\( T^{12} - T^{11} + 12 T^{10} + \cdots + 4431025 \)
T^12 - T^11 + 12*T^10 + 32*T^9 + 545*T^8 + 2538*T^7 + 16207*T^6 + 62141*T^5 + 255601*T^4 + 616635*T^3 + 1363885*T^2 + 2210250*T + 4431025
$31$
\( T^{12} - 13 T^{11} + 44 T^{10} + \cdots + 8410000 \)
T^12 - 13*T^11 + 44*T^10 + 263*T^9 + 2351*T^8 - 35605*T^7 + 179220*T^6 - 513775*T^5 + 5500025*T^4 - 7774000*T^3 + 15808000*T^2 - 9425000*T + 8410000
$37$
\( T^{12} - 8 T^{11} + 44 T^{10} + \cdots + 36300625 \)
T^12 - 8*T^11 + 44*T^10 - 222*T^9 + 2576*T^8 - 1550*T^7 + 18455*T^6 - 101000*T^5 + 867525*T^4 + 4007250*T^3 + 26472625*T^2 + 23196250*T + 36300625
$41$
\( T^{12} + 8 T^{11} + \cdots + 6831849025 \)
T^12 + 8*T^11 + 179*T^10 + 1537*T^9 + 17636*T^8 + 85585*T^7 + 638380*T^6 + 1803925*T^5 + 43623460*T^4 + 270723125*T^3 + 1749002375*T^2 + 5018398325*T + 6831849025
$43$
\( (T^{6} + 2 T^{5} - 91 T^{4} - 174 T^{3} + \cdots - 6284)^{2} \)
(T^6 + 2*T^5 - 91*T^4 - 174*T^3 + 1369*T^2 + 1422*T - 6284)^2
$47$
\( T^{12} - 13 T^{11} + 178 T^{10} + \cdots + 5216656 \)
T^12 - 13*T^11 + 178*T^10 - 889*T^9 + 4013*T^8 + 9289*T^7 - 14118*T^6 + 228711*T^5 + 3541003*T^4 + 10666294*T^3 + 15308248*T^2 + 9346128*T + 5216656
$53$
\( T^{12} + 44 T^{11} + \cdots + 1189905025 \)
T^12 + 44*T^11 + 976*T^10 + 13394*T^9 + 128756*T^8 + 872810*T^7 + 4409895*T^6 + 17567060*T^5 + 75162585*T^4 + 178298250*T^3 + 369144525*T^2 + 683345950*T + 1189905025
$59$
\( T^{12} - 22 T^{11} + 213 T^{10} + \cdots + 15366400 \)
T^12 - 22*T^11 + 213*T^10 - 544*T^9 + 2225*T^8 + 10236*T^7 + 108283*T^6 - 608092*T^5 + 1826161*T^4 + 1218420*T^3 + 13120240*T^2 + 8232000*T + 15366400
$61$
\( T^{12} + 8 T^{11} + \cdots + 28314456361 \)
T^12 + 8*T^11 + 53*T^10 + 289*T^9 + 11178*T^8 + 82331*T^7 + 1252612*T^6 + 9010629*T^5 + 83532878*T^4 + 415809961*T^3 + 2125615763*T^2 + 2319251627*T + 28314456361
$67$
\( T^{12} + 6 T^{11} + 29 T^{10} + \cdots + 215619856 \)
T^12 + 6*T^11 + 29*T^10 - 590*T^9 + 8600*T^8 + 16796*T^7 + 1493906*T^6 - 5374286*T^5 + 27556405*T^4 - 51076650*T^3 + 98258104*T^2 - 151303936*T + 215619856
$71$
\( T^{12} - 21 T^{11} + 259 T^{10} + \cdots + 38416 \)
T^12 - 21*T^11 + 259*T^10 - 1125*T^9 + 4950*T^8 + 71639*T^7 + 691931*T^6 + 17926*T^5 + 14511625*T^4 + 16339400*T^3 + 6976424*T^2 - 249704*T + 38416
$73$
\( T^{12} + 16 T^{11} + \cdots + 493062025 \)
T^12 + 16*T^11 + 191*T^10 + 881*T^9 + 2311*T^8 + 50305*T^7 + 1271285*T^6 + 10361630*T^5 + 51790035*T^4 + 165631950*T^3 + 377325100*T^2 + 484624125*T + 493062025
$79$
\( T^{12} - 10 T^{11} - 35 T^{10} + \cdots + 64000000 \)
T^12 - 10*T^11 - 35*T^10 + 865*T^9 + 17860*T^8 - 138925*T^7 + 730875*T^6 - 2679425*T^5 + 10174025*T^4 - 22452000*T^3 + 53360000*T^2 - 52800000*T + 64000000
$83$
\( T^{12} - 10 T^{11} + \cdots + 1683953296 \)
T^12 - 10*T^11 + 217*T^10 - 2734*T^9 + 29189*T^8 - 166606*T^7 + 924089*T^6 - 1526378*T^5 + 10885437*T^4 - 18676802*T^3 + 10439084*T^2 + 150109688*T + 1683953296
$89$
\( T^{12} + 57 T^{11} + \cdots + 142170473025 \)
T^12 + 57*T^11 + 1683*T^10 + 29914*T^9 + 353250*T^8 + 2361589*T^7 + 7501473*T^6 + 1696647*T^5 + 749196316*T^4 + 5534078130*T^3 + 70721566965*T^2 - 51903506025*T + 142170473025
$97$
\( T^{12} - 4 T^{11} + 139 T^{10} + \cdots + 5755201 \)
T^12 - 4*T^11 + 139*T^10 - 355*T^9 + 16020*T^8 - 50839*T^7 + 1982626*T^6 - 6104141*T^5 + 202353010*T^4 - 564755965*T^3 + 6155269049*T^2 + 304699389*T + 5755201
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